Representation of a Lie Group

Let $G$ be a Lie group and $V$ a finite-dimensional vector space . A (linear) representation of $G$ is a Lie group homomorphism

where $\operatorname{GL}(V)$ is the group of invertible linear operators on $V$.

Equivalently, $\rho$ is a smooth action of $G$ on $V$ by linear isomorphisms, written $g\cdot v := \rho(g)v$.

Differentiating a representation

The differential at the identity gives a Lie algebra representation

which is a representation of the Lie algebra $\mathfrak{g}=T_eG$; see Lie algebra of a Lie group and differential .

Examples

• The defining representation $\operatorname{GL}(n,\mathbb{R})\to \operatorname{GL}(\mathbb{R}^n)$.
• Orthogonal/unitary representations of $\operatorname{SO}(n)$, $\operatorname{SU}(n)$ on $\mathbb{R}^n$, $\mathbb{C}^n$.
• The adjoint representation $G\to \operatorname{Aut}(\mathfrak{g})$.

Many structural notions for representations can be studied via the induced Lie algebra representation and tools such as the Killing form in the semisimple case.